7.1.2. Introduction to Navier-Stokes Equations

Today we are going to discuss very interesting topics, the topics on Navier-Stokes equations. They form the basic foundations of computational fluid dynamics and help solve many complex fluid flow problems.

The Navier-Stokes equations describe how fluids move. They are fundamental to the field of fluid dynamics and are essential for understanding various fluid flow scenarios, from weather patterns to the flow of air over wings. These equations consider factors like velocity, pressure, density, and external forces acting on the fluid, providing a comprehensive description of fluid behavior.

Think of the Navier-Stokes equations as the recipe for a complex dish, like a soufflé. Just as various ingredients and their proportions influence the final dish, factors like velocity and pressure interact to influence how a fluid will behave in various situations.

The Navier-Stokes equations were derived independently by French physicist Claude-Louis Navier and English mathematician George Stokes, both around 200 years ago.

Claude-Louis Navier and George Stokes contributed significantly to fluid mechanics. Their independent work led to the derivation of the Navier-Stokes equations, allowing for the mathematical modeling of fluid motion. The collaboration of a physicist and a mathematician highlights the importance of interdisciplinary approaches in science.

Imagine two chefs working in separate kitchens but creating very similar dishes based on the same ingredient lists. Later, they discover each other's recipes and realize that they both figured out how to achieve the same delicious outcome through different methods.

I will also talk about revisiting Newtonian fluid and non-Newtonian fluid parts, which are important to look at how we can use for this Navier-Stokes equations.

Newtonian fluids have a constant viscosity, meaning their flow behavior remains consistent regardless of the force applied to them. Examples include water and air. In contrast, non-Newtonian fluids display variable viscosity depending on the stress or rate of shear applied. A common example is ketchup, which flows easily when shaken but remains thick when left undisturbed. Understanding these differences is crucial for correctly applying the Navier-Stokes equations.

Consider a well-mixed smoothie (Newtonian fluid) that pours smoothly compared to honey (non-Newtonian fluid), which flows slowly and requires more force to move. The transition from thick to thin flow showcases how different forces affect fluid dynamics.

We will talk about the derivations of Navier-Stokes equations with assumptions of incompressible isothermal flow.

The Navier-Stokes equations can be derived by applying the principles of conservation of momentum and mass, along with the assumptions of incompressibility and constant temperature (isothermal). These assumptions greatly simplify the equations, allowing for clearer analysis of fluid dynamics. The derivation process involves breaking down forces acting on a fluid element, including pressure gradients and viscosity, leading to the final form of the equations.

Imagine trying to understand how water flows through a garden hose. By assuming the water's temperature remains constant and that it doesn't compress under pressure, you simplify the problem. In this analogy, the simplifications help clarify how the water flows — demonstrating the value of assumptions in complex calculations.

We will talk about how we can form the Navier-Stokes equations in two coordinate systems – one is the Cartesian coordinate system and one is the cylindrical coordinate system.

The Navier-Stokes equations can be expressed in different coordinate systems, such as Cartesian or cylindrical coordinates. The choice of coordinate system can simplify the equations based on the problem's geometry. For instance, cylindrical coordinates are particularly useful for problems involving rotation or symmetrical flow around a cylinder. Understanding these coordinate transformations helps in solving fluid dynamics problems more efficiently.

Think of it like using a map to navigate. If you're traveling through the city, a street map (Cartesian coordinates) is helpful, but if you're navigating around a circular park, a park map (cylindrical coordinates) would be more effective. The right tools simplify your journey.

After discussing the derivations, we can look at example problems with appropriate boundary conditions to solve the Navier-Stokes equations.

Once the Navier-Stokes equations are established, applying them to real-world problems involves specifying boundary conditions — constraints that must be met within the fluid domain. For example, in a pipe flow situation, the fluid's speed at the pipe walls might be zero (no-slip condition), affecting how the equations are solved. Understanding these boundary conditions is essential for deriving meaningful and accurate solutions.

Imagine trying to fill a balloon (the fluid domain) with air (the fluid) while ensuring it doesn’t burst. The condition that the balloon's surface must remain intact is akin to a boundary condition placed on the Navier-Stokes equations. It sets the stage for how the air (fluid) behaves inside.

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